Space–time generalized finite difference scheme for three-dimensional unsteady nonlinear convection–diffusion–reaction equation

This paper develops a space–time generalized finite difference method (ST-GFDM), integrated with a time-marching framework and the Levenberg–Marquardt algorithm (LMA), to address three-dimensional unsteady nonlinear convection–diffusion–reaction equations. The ST-GFDM, as a meshless approach combined with a space–time formulation, approximates spatial and temporal derivatives by solving a local least-squares system derived from Taylor series expansion within the ST domain. Through this formulation, the governing PDEs are converted into a nonlinear algebraic system, efficiently resolved via a two-step LMA iteration. The time-marching mechanism incrementally propagates the ST computational domain forward along the temporal axis, which significantly reduces memory usage and enhances performance in long-duration simulations. The unified treatment of time and space discretization further improves numerical robustness, alleviating sensitivity to parameters that typically challenge conventional solvers in high-dimensional transient problems. Two benchmark tests are conducted to validate the effectiveness and applicability of the proposed meshless framework for solving 3D nonlinear convection–diffusion–reaction systems.